Home > APCALC > Chapter 9 > Lesson 9.1.3 > Problem9-45

9-45.

A variant of Ying’s method (manipulating an infinite sum so that the sum appears as a part of itself) can be used in other situations. For example, to evaluate the infinite “nested radical” below, let:

$S = \sqrt { 2 + \sqrt { 2 + \sqrt { 2 + \sqrt { 2 + \ldots } } } }$

Explain why  $S =\sqrt { 2 + S }$. Then solve for $S$.

$S^2=2+S$

$S^2-S-2=0$

$(S-2)(S+1)=0$

$S-2=0\text{ or }S+1=0$

Which sum makes sense?